Positive Borel measures

This is a note of real and complex analysis chapter 2.

Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set ($C$) ($\Lambda f$) represents the integration of the function ($\int fdu$) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let $\Lambda$ be a positive linear functional on $C_c\left(X\right)$. Then there exist a $\sigma -algebra$ in $X$ which contains all Borel sets in $X$, and there exists a unique positive measure $mu$ on $M$ which represents $\Lambda$ in the sense that (a) $\Lambda f=\int fd\mu$ for every $f\in C_c\left(X\right)$ and following additional properties:
(b) $\mu \left(K\right)<\infty$ for every compact set $K\subset X$.
(c) For every $E\in M$, we have $$\mu \left(E\right)=inf\left\{\mu \right(V):EinV,Vopen\}$$.
(d) The relation $$\mu \left(E\right)=sup\left\{\mu \right(K):K\in E,Kcompact\}$$
holds for every open set $E$, and for every $E\in M$ with $\mu \left(E\right)<\infty$.
(e) If $E\in M,AsubsetE$, and $\mu \left(E\right)=0$, then $A\in M$.

The Riesz theorem is about linear functional $\Lambda$ is equivalently replaced with choosing measure $\mu \left(E\right)=sup\{\Lambda f:f\prec V\}$. Note $sup\left\{{\int }_0^{1}f\right(x)dx=\Lambda f:f\prec V,V(0,1\left)\right\}=1$. The notion of $\prec$ include $0\le f\le 1$.

I confused $C_c\left(X\right)$